Question

How do you verify the identity `sin(x+y)sin(x-y)=sin^2x-sin^2y`?

Answer 1

Consider the trig identities:
sin (x + y) = sin x.cos y + sin y.cos x
sin (x - y) = sin x.cos y - sin y.cos x
Applying the algebraic identity: `(a + b)(a - b) = a^2- b^2`, their
product becomes:
`P = sin (x + y).sin (x - y) = sin^2 x.cos^2 y - sin^2 y.cos^2 x`
Replace `cos^2 y` by `(1 - sin^2 y)` and replace
`cos^2 x` by `(1 - sin^2 x)`.
We get:
`P = sin^2 x - sin^2 x.sin^2 y - sin^2 y + sin^2 y.sin^2 x`
`P = sin^2 x - sin^2 y`